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The development of classical ergodic theory has had a significant impact inthe areas of mathematics, physics, and, in general, applied sciences. Thequantum ergodic theory of Hamiltonian dynamics has its motivations tounderstand thermodynamics and statistical mechanics and is still debated a lot.Quantum channel, a completely positive trace-preserving map, represents a mostgeneral representation of quantum dynamics and is an essential aspect ofquantum information theory and quantum computation. In this work, we study theergodic theory of quantum channels by characterizing different levels ofergodic hierarchy from integrable to mixing. The quantum channels on singlesystems are constructed from the unitary operators acting on bipartite statesand tracing out the environment. The interaction strength of these unitaryoperators measured in terms of operator entanglement provides sufficientconditions for the channel to be mixing. By using block diagonal unitaryoperators, we construct a set of non-ergodic channels. From integrable tomixing is characterized explicitly in the case of the two-qubit unitaryoperator. Moreover, we also study interacting many-body quantum systems thatinclude the famous Sachdev-Ye-Kitaev (SYK) model and show that they displaymixing within the framework of the quantum channel.

Ergodic and mixing quantum channels: From two-qubit to many-body quantum systems